1 Source-Channel Coding for Fading Channels with Correlated …communications/Papers/fdpc-tcom.pdf · 2014. 9. 26. · 1 Source-Channel Coding for Fading Channels with Correlated

1

Source-Channel Coding for Fading Channels

with Correlated Interference

Ahmad Abou Saleh, Wai-Yip Chan, and Fady Alajaji

Abstract

We consider the problem of sending a Gaussian source over a fading channel with Gaussian

interference known to the transmitter. We study joint source-channel coding schemes for the case of

unequal bandwidth between the source and the channel and when the source and the interference are

correlated. An outer bound on the system’s distortion is first derived by assuming additional information

at the decoder side. We then propose layered coding schemes based on proper combination of power

splitting, bandwidth splitting, Wyner-Ziv and hybrid coding. More precisely, a hybrid layer, that uses

the source and the interference, is concatenated (superimposed) with a purely digital layer to achieve

bandwidth expansion (reduction). The achievable (square error) distortion region of these schemes under

matched and mismatched noise levels is then analyzed. Numerical results show that the proposed

schemes perform close to the best derived bound and to be resilient to channel noise mismatch. As

an application of the proposed schemes, we derive both inner and outer bounds on the source-channel-

state distortion region for the fading channel with correlated interference; the receiver, in this case, aims

to jointly estimate both the source signal as well as the channel-state (interference).

Index Terms

Joint source-channel coding, distortion region, correlated interference, dirty paper coding, hybrid

digital-analog coding, fading channels.

This work was supported in part by NSERC of Canada.

A. Abou Saleh and W-Y. Chan are with the Department of Electrical and Computer Engineering, Queen’s University, Kingston,

ON, K7L 3N6, Canada (e-mail: [email protected]; [email protected]).

F. Alajaji is with the Department of Mathematics and Statistics, Queen’s University, Kingston, ON, K7L 3N6, Canada (e-mail:

[email protected]).

September 12, 2014 DRAFT

2

I. INTRODUCTION

The traditional approach for analog source transmission in point-to-point communications

systems is to employ separate source and channel coders. This separation is (asymptotically)

optimal given unlimited delay and complexity in the coders [1]. There are, however, two disad-

vantages associated with digital transmission. One is the threshold effect: the system typically

performs well at the design noise level, while its performance degrades drastically when the

true noise level is higher than the design level. This effect is due to the quantizers sensitivity

to channel errors and the eventual breakdown of the employed error correcting code at high

noise levels (no matter how powerful it is). The other trait is the levelling-off effect: as the noise

level decreases, the performance remains constant beyond a certain threshold. This is due to the

non-recoverable distortion introduced by the quantizer which limits the system performance at

low noise levels. Joint source-channel coding (JSCC) schemes are more robust to noise level

mismatch than tandem systems which use separate source and channel coding. Analog JSCC

schemes are studied in [2]–[11]. These schemes are based on the so-called direct source-channel

mappings. A family of hybrid digital-analog (HDA) schemes are introduced in [12]–[14] to

overcome the threshold and the levelling-off effects. In [15]–[17], HDA schemes are proposed

for broadcast channels and Wyner-Ziv systems.

It is well known that for the problem of transmitting a Gaussian source over an additive white

Gaussian noise (AWGN) channel with interference that is known to the transmitter, a tandem

Costa coding scheme, which comprises an optimal source encoder followed by Costa’s dirty

paper channel code [18], is optimal in the absence of correlation between the source and the

interference. In [19], the authors studied the same problem as in [18] and proposed an HDA

scheme (for the matched bandwidth case) that is able to achieve the optimal performance (same

as the tandem Costa scheme). In [20], the authors adapted the scheme proposed in [19] for

the bandwidth reduction case. In [21], the authors proposed an HDA scheme for broadcasting

correlated sources and showed that their scheme is optimal whenever the uncoded scheme

of [22] is not. In [23], the authors studied HDA schemes for broadcasting correlated sources

under mismatched source-channel bandwidth; in [24], the authors studied the same problem

and proposed a tandem scheme based on successive coding. In [25], we derived inner and outer

bounds on the system’s distortion for the broadcast channel with correlated interference. Recently,


3

[26] studied a joint source channel coding scheme for transmitting analog Gaussian source over

AWGN channel with interference known to the transmitter and correlated with the source. The

authors proposed two schemes for the matched source-channel bandwidth; the first one is the

superposition of the uncoded signal and a digital signal resulting from the concatenation of a

Wyner-Ziv coder [27] and a Costa coder, while in the second scheme the digital part is replaced

by an HDA part proposed in [19]. In [28], we consider the problem of [26] under bandwidth

expansion; more precisely, we studied both low and high-delay JSCC schemes. The limiting

case of this problem, where the source and the interference are fully correlated was studied

in [29]; the authors showed that a purely analog scheme (uncoded) is optimal. Moreover, they

also considered the problem of sending a digital (finite alphabet) source in the presence of

interference where the interference is independent from the source. More precisely, the optimal

tradeoff between the achievable rate for transmitting the digital source and the distortion in

estimating the interference is studied; they showed that the optimal rate-state-distortion tradeoff

is achieved by a coding scheme that uses a portion of the power to amplify the interference and

uses the remaining power to transmit the digital source via Costa coding. In [30], the authors

considered the same problem as in [29] but with imperfect knowledge of the interference at the

transmitter side.

In this work, we study the reliable transmission of a memoryless Gaussian source over a

Rayleigh fading channel with known correlated interference at the transmitter. More precisely,

we consider equal and unequal source-channel bandwidths and analyze the achievable distortion

region under matched and mismatched noise levels. We propose a layered scheme based on hybrid

coding. One application of JSCC with correlated interference can be found in sensor network

and cognitive radio channels where two nodes interfere with each other. One node transmits

directly its signal; the other, however, is able to detect its neighbour node transmission and treat

it as a correlated interference. In [31], we studied this problem under low-delay constraints;

more specifically, we designed low-delay source-channel mappings based on joint optimization

between the encoder and the decoder. One interesting application of this problem is to study

the source-channel-state distortion region for the fading channel with correlated interference;

in that case, the receiver side is interested in estimating both the source and the channel-state

(interference). Inner and outer bounds on the source-interference distortion region are established.

Our setting contains several interesting limiting cases. In the absence of fading and for the


4

matched source-channel bandwidth, our system reverts to that of [26]; for the uncorrelated source-

interference scenario without fading, our problem reduces to the one in [20] for the bandwidth

reduction case. Moreover, the source-channel-state transmission scenario generalizes the setting

in [29] to include fading and correlation between source and interference. The rest of the paper

is organized as follows. In Section II, we present the problem formulation. In Section III, we

derive an outer bound and introduce linear and tandem digital schemes. In Section IV, we

derive inner bounds (achievable distortion region) under both matched and mismatched noise

levels by proposing layered hybrid coding schemes. We extend these inner and outer bounds to

the source-channel-state communication scenario in Section V. Finally, conclusions are drawn

in Section VI.

Throughout the paper, we will use the following notation. Vectors are denoted by characters

superscripted by their dimensions. For a given vector XN = (X(1), ..., X(N))T , we let [XN ]K1and [XN ]NK+1 denote the sub-vectors [X

N ]K1 , (X(1), ..., X(K))T and [XN ]NK+1 , (X(K +

1), ..., X(N))T , respectively, where (·)T is the transpose operator. When there is no confusion,

we also write [XN ]K1 as XK . When all samples in a vector are independent and identically

distributed (i.i.d.), we drop the indexing when referring to a sample in a vector (i.e., X(i) = X).

II. PROBLEM FORMULATION AND MAIN CONTRIBUTIONS

We consider the transmission of a Gaussian source V K = (V (1), ..., V (K))T ∈ RK over

a Rayleigh fading channel in the presence of Gaussian interference SN ∈ RN known at the

transmitter (see Fig. 1). The source vector V K represents the first K samples of V max (K,N); SN

is similarly defined. The source vector V K , which is composed of i.i.d. samples, is transformed

into an N dimensional channel input XN ∈ RN using a nonlinear mapping function, in general,

α(.) : RK × RN → RN . The received symbol is Y N = FN(XN + SN) + WN , where addition

and multiplication are component-wise, FN represents an N -block Rayleigh fading that is

independent of (V K ;SN ;WN) and known to the receiver side only, XN = α(V K , SN), SN

is an i.i.d. Gaussian interference vector (with each sample S ∼ N (0, σ2S)) that is considered to

be the output of a side channel with input V max (K,N) as shown in Fig. 1, and each sample in the

additive noise WN is drawn from a Gaussian distribution (W ∼ N (0, σ2W )) independently from

both the source and the interference. Unlike the typical dirty paper problem which assumes

an AWGN channel with interference (that is uncorrelated to the source) [18], we consider a


5

fading channel and assume that V K and SN are jointly Gaussian. Since the fading realization

is known only at the receiver, we have partial knowledge of the actual interference FNSN at

the transmitter. In this work, we assume that only V (i) and S(i), i = 1, ...,min (K,N), are

correlated according to the following covariance matrix

ΣV S =

σ2V ρσV σSρσV σS σ

2S

(1)where σ2V , σ

2S are, respectively, the variance of the source and the interference, and ρ is the source-

interference correlation coefficient. The system operates under an average power constraint P

E[||α(V K , SN)||2]/N ≤ P (2)

where E[(·)] denotes the expectation operator. The reconstructed signal is given by V̂ K =

γ(Y N , FN), where the decoder is a mapping from RN × RN → RK . The rate of the system is

given by r = NK

channel use/source symbol. When r = 1, the system has an equal-bandwidth

between the source and the channel. For r < 1 (r > 1), the system performs bandwidth reduction

(expansion). According to the correlation model described above, note that for r < 1, the first

N source samples [V K ]N1 and SN are correlated via the covariance matrix in (1), while the

remaining K − N samples [V K ]KN+1 and SN are independent. For r > 1, however, V K and

[SN ]K1 are correlated via the covariance matrix in (1), while VK and [SN ]NK+1 are uncorrelated.

V K

SN

α(.) +

WN

XN Y N γ(.)V̂ K

+ x

FN

V max(K,N) SideChannel

Smax(K,N)

Fig. 1. A K : N system structure over a fading channel with interference known at the transmitter side. The interference

Smax (K,N) is assumed to be the output of a noisy side channel with input V max (K,N). V K represents the first K samples of

V max (K,N) (SN is defined similarly). The fading coefficient is assumed to be known at the receiver side; the transmitter side,

however, knows the fading distribution only.

In this paper, we aim to find a source-channel encoder α and decoder γ that minimize the mean

square error (MSE) distortion D = E[||V K − V̂ K ||2]/K under the average power constraint in

(2). For a particular coding scheme (α, γ), the performance is determined by the channel power

constraint P , the fading distribution, the system rate r, and the incurred distortion D at the


6

receiver. For a given power constraint P , fading distribution and rate r, the distortion region is

defined as the closure of all distortions Do for which (P,Do) is achievable. A power-distortion

pair is achievable if for any δ > 0, there exist sufficiently large integers K and N with N/K = r,

a pair of encoding and decoding functions (α, γ) satisfying (2), such that D < Do + δ. In this

work, we analyze the distortion for equal and unequal bandwidths between the source and the

channel with no constraint on the delay (i.e., both N and K tend to infinity with NK

= r fixed).

Our main contributions can be summarized as follows

• We derive inner and outer bounds for the system’s distortion region for a Gaussian source

over fading channel with correlated interference under equal and unequal source-channel

bandwidths. The outer bounds are found by assuming full/partial knowledge of the inter-

ference at the decoder side. The inner bounds are derived by proposing hybrid coding

schemes and analyzing their achievable distortion region. These schemes are based on

proper combination of power splitting, bandwidth splitting, Wyner-Ziv and hybrid coding; a

hybrid layer that uses the source and the interference is concatenated (superimposed) with a

purely digital layer to achieve bandwidth expansion (reduction). Different from the problem

considered in [26], we consider the case of fading and mismatch in the source-channel

bandwidth. Our scheme offers better performance than the one in [26] under matched

bandwidth (when accommodating the Costa coder in their scheme for fading channels).

Moreover, our scheme is optimal when there is no fading and when the source-interference

are either uncorrelated or fully correlated.

• As an application of the proposed schemes, we consider source-channel-state transmission

over fading channels with correlated interference. In such case, the receiver aims to jointly

estimate both the source signal as well as the channel-state. Inner and outer bounds are

derived for this scenario. For the special case of uncorrelated source-interference over

AWGN channels, we obtain the optimal source-channel-state distortion tradeoff; this result

is analogous to the optimal rate-state distortion for the transmission of a finite discrete source

over a Gaussian state interference derived in [29]. For correlated source-interference and

fading channels, our inner bound performs close to the derived outer bound and outperforms

the adapted scheme of [29].


7

III. OUTER BOUNDS AND REFERENCE SYSTEMS

A. Outer Bounds

In [26] and [32], outer bounds on the achievable distortion were derived for point-to-point

communication over Gaussian channel with correlated interference under matched bandwidth

between the source and the channel. This was done by assuming full/partial knowledge of the

interference at the decoder side. In this section, for the correlation model considered above, we

derive outer bounds for the fading interference channel under unequal source-channel bandwidth.

Since S(i) and V (i) are correlated for i = 1, ...,min (K,N), we have S(i) = SI(i) + SD(i),

with SD(i) = ρσSσV V (i) and SI ∼ N (0, (1− ρ2)σ2S) are independent of each other. To derive an

outer bound, we assume knowledge of both (S̃K , [SN ]NK+1) and FN at the decoder side for the

case of bandwidth expansion, where S̃K = η1SKI +η2SKD (the linear combination S̃ is motivated

by [32]), and (η1, η2) is a pair of real parameters. For the bandwidth reduction case, we assume

knowledge of S̃N and FN at the decoder to derive a bound on the average distortion for the

first N samples; the derivation of a bound on the average distortion for the remaining K − N

samples assumes knowledge of [V K ]N1 in addition to S̃N .

Definition 1 Let MSE(Y ; S̃) be the distortion incurred from estimating Y based on S̃ using

a linear minimum MSE estimator (LMMSE) denoted by γlmse(S̃K , fK). This distortion, which

is a function of η1, η2, E[XSI ] and E[XSD], is given by MSE(Y ; S̃) = E[(Y −γlmse(S̃K , fK))2] =(E[Y 2]− (E[Y S̃])

2

E[S̃2]

), where E[Y 2] = f 2(P+σ2S+2(E[XSI+XSD]))+σ2W , E[Y S̃] = f

(E[X(η1SI+

η2SD)] +E[η1S2I + η2S2D])

and E[S̃2] = E[η21S2I + η22S2D]. These terms will be used in Lemmas 1

and 2.

Lemma 1 For a K : N bandwidth expansion system with N ≥ K (the matched case is treated

as a special case), the outer bound on the system’s distortion D can be expressed as follows

D ≥ Dob , supη1,η2

infX:

|E[XSI ]|≤√

E[X2]E[S2I ]

|E[XSD]|≤√

E[X2]E[S2D]

Var(V |S̃)

exp{EF[log

((MSE(Y ;S̃)

σ2W

)(f2P+σ2W

σ2W

)r−1)]} (3)

where Var(V |S̃) = σ2V(1− η

22ρ

2

η21(1−ρ2)+η22ρ2)

is the variance of V given S̃.


8

Proof: For a K : N system with N ≥ K, we have the following

K

2log

Var(V |S̃)D

≤ I(V K ; V̂ K |S̃K , [SN ]NK+1, FN) ≤ I(V K ;Y N |S̃K , [SN ]NK+1, FN)

= h(Y N |S̃K , [SN ]NK+1, FN)− h(Y N |V K , SN , FN)

≤ h(Y K |S̃K , FK) + h([Y N ]NK+1∣∣[SN ]NK+1, [FN ]NK+1)− h(Y N |V K , SN , FN)

= EF[h(Y K |S̃K , fk) + h([Y N ]NK+1

∣∣[SN ]NK+1, [fn]nk+1)]− h(WN)≤ EF

[K

2log 2πe(MSE(Y ; S̃)) +

N −K2

log 2πe(f 2P + σ2W )

]− N

2log 2πeσ2W

= EF

[K

2log

(MSE(Y ; S̃)

σ2W

)+N −K

2log

(f 2P + σ2W

σ2W

)](4)

where we used h(Y K |S̃K , fK) ≤ h(Y K − γlmse(S̃K , fK)) ≤ K2 log 2πe(

MSE(Y ; S̃))

. By the

Cauchy-Schwarz inequality, we have |E[XSI ]| ≤√E[X2]E[S2I ] and |E[XSD]| ≤

√E[X2]E[S2D].

For a given η1 and η2, we have to choose the highest value of MSE(Y ; S̃) over E[XSD] and

E[XSI ]; then we need to maximize the right-hand side of (3) over η1 and η2. Note that most

inequalities follow from rate-distortion theory, the data processing inequality and the facts that

conditioning reduces differential entropy and that the Gaussian distribution maximizes differential

entropy.

Lemma 2 For K : N bandwidth reduction (K > N ), the outer bound on D is given by

D ≥ Dob(ξ∗) , supη1,η2

infξ

infX:

|E[XSI ]|≤√

(1−ξ)PE[S2I ]|E[XSD]|≤

√(1−ξ)PE[S2D]

r Var(V |S̃)exp{EF [log ( MSE(Y ;S̃)ξPf2+σ2W )]}

+(1− r) σ2V

exp{EF[

NK−N log

(ξPf2+σ2W

σ2W

)]} (5)

where ξ ∈ [0, 1].

Proof: We start by decomposing the average MSE distortion as follows

D =1

KE[||V K − V̂ K ||2] = 1

K

(E[||V N − V̂ N ||2] + E[||[V K ]KN+1 − [V̂ K ]KN+1||2]

)=

N

K

(1

NE[||V N − V̂ N ||2]

)+K −NK

(1

K −NE[||[V K ]KN+1 − [V̂ K ]KN+1||2]

)= rD1 + (1− r)D2 (6)


9

where D1 and D2 are the average distortion in reconstructing V N and [V K ]KN+1, respectively. To

find an outer bound on D, we derive bounds on both D1 and D2. To bound D1, We can write

the following expressionN

2log

Var(V |S̃)D1

≤ I(V N ; V̂ N |S̃N , FN) ≤ I(V N ;Y N |S̃N , FN)

= h(Y N |S̃N , FN)− h(Y N |S̃N , V N , FN)

= h(Y N |S̃N , FN)− h(Y N |SN , V N , FN)(a)

≤ EF[N

2log 2πe(MSE(Y ; S̃))− N

2log 2πe(ξPf 2 + σ2W )

]≤ sup

Y ∈AEF

[N

2log

(MSE(Y ; S̃)ξPf 2 + σ2W

)](7)

where the set A = {Y : h(Y N |SN , V N , FN) = EF[N2

log 2πe(ξPf 2 + σ2W )]}. Note that in (7)-

(a) we use the fact that h(Y N |SN , V N , FN) = EF[N2

log 2πe (ξPf 2 + σ2W )], for some ξ ∈ [0 1].

This can be shown by noting that the following inequality holds N2

log 2πe(σ2W ) = h(WN) ≤

h(Y N |SN , V N , FN) ≤ h(FNXN +WN |FN) = EF [N2 log 2πe(Pf2 + σ2W )]; as a result, there is

a ξ ∈ [0 1] such that h(Y N |SN , V N , FN) = EF[N2

log 2πe (ξPf 2 + σ2W )]. Moreover in (7)-(a),

we used the fact that

h(Y N |S̃N , FN) = EF [h(Y N |S̃N , fn)] = EF [h(Y N − γlmse(S̃N , fn)|S̃N , fn)]

≤ EF [h(Y N − γlmse(S̃N , fn))] ≤N

2EF [log 2πe(MSE(Y ; S̃))]. (8)

Similarly, to derive a bound on D2, we have the followingK −N

2log

σ2VD2

≤ I([V K ]KN+1; [V̂ K ]KN+1|SN , V N , FN) ≤ I([V K ]KN+1;Y N |SN , V N , FN)

= h(Y N |SN , V N , FN)− h(Y N |SN , V N , [V K ]KN+1, FN)

= h(Y N |SN , V N , FN)− h(Y N |SN , V K , FN)

= E[N

2log

(ξPf 2 + σ2W

σ2W

)](9)

where in the last equality, we used h(Y N |SN , V N , FN) = EF[N2

log 2πe (ξPf 2 + σ2W )]

as

shown earlier. Note that since we do not know the value of ξ, the overall distortion has to

be minimized over the parameter ξ. Now using (7) and (9) in (6), we have the following bound

D ≥ infξ

infY ∈A

r Var(V |S̃)exp{EF [log (MSE(Y ;S̃)ξPf2+σ2W )]} + (1− r)σ2V

exp{EF[

NK−N log

(ξPf2+σ2W

σ2W

)]}(10)


10

where the sup in (7) is manifested as inf on the distortion. Note that the above sequence of

inequalities in (8) becomes equalities when Y is conditionally Gaussian given F and when

Y − γlmse(S̃, f) and S̃ are jointly Gaussian and orthogonal to each other given F ; this happens

when X∗ is jointly Gaussian with S, V and W given F . Hence, the sup in (7) happens when

X∗ is Gaussian. Now we write X∗ = N∗ξ +X∗ξ , where N

∗ξ ∼ N (0, ξP ) is independent of (V, S)

and X∗ξ ∼ N (0, (1 − ξ)P ) is a function of (V, S). Note that X∗ξ is independent of N∗ξ . As a

result , the equality h(Y N |SN , V N , FN) = EF[N2

log 2πe (ξPf 2 + σ2W )]

still holds and hence

Y ∗ ∈ A, E[Y 2] = f 2(P+σ2S+2(E[X∗ξSI+X∗ξSD]))+σ2W and E[Y S̃] = f(E[X∗ξ (η1SI+η2SD)]+

E[η1S2I +η2S2D]). By the Cauchy-Schwarz inequality, |E[X∗SI ]| = |E[X∗ξSI ]| ≤

√E[(X∗ξ )2]E[S2I ]

and |E[X∗SD]| = |E[X∗ξSD]| ≤√

E[(X∗ξ )2]E[S2D]. Hence we maximize the value of MSE(Y ; S̃)

over X or equivalently over E[X∗ξSI ] and E[X∗ξSD] satisfying the above constraints. Finally, the

parameters η1 and η2 are chosen so that the right hand side of (5) is maximized.

B. Linear Scheme

In this section, we assume that the encoder transforms the K-dimensional signal V K into an

N -dimensional channel input XN using a linear transformation according to

XN = α(V K , SN) = TV K + MSN (11)

where T and M are RN×K and RN×N matrices, respectively. In such case, Y N is condi-

tionally Gaussian given FN and the minimum MSE (MMSE) decoder is a linear estimator,

with, V̂ K = ΣV Y Σ−1Y YN , where ΣV Y = E

[(V K)(Y N)T

]and ΣY = E

[(Y N)(Y N)T

]. The

matrices T and M can be found (numerically) by minimizing the MSE distortion Dlinear =

EF[

1Ktr{σ2V IK×K − ΣV Y Σ−1Y ΣTV Y

}]under the power constraint in (2), where tr(.) is the trace

operator and IK×K is a K ×K identity matrix. Note that by setting M to be the zero matrix

and T =√P/σ2V IN×K , the system reduces to the uncoded scheme. Focusing on the matched

case (K = N), we have the following lemma for finite block length K.

Lemma 3 For the matched-bandwidth source-channel coding of a Gaussian source transmitted

over an AWGN fading channel with correlated interference, the distortion lower bound for any

linear scheme is achieved with single-letter linear codes.


11

Proof: Recall that since V K and SK are correlated, we have SK = ρσSσVV K +NKρ , where the

samples in NKρ are i.i.d. Gaussian with common variance σ2S(1−ρ2). As a result and using (11)

Y K = F

(T +

ρσSσV

M +ρσSσV

IK×K

)V K + F (M + IK×K)N

Kρ +W

K

= FT̃V K + FM̃NKρ +WK (12)

where F = diag(FK) is a diagonal matrix that represents the fading channel, M̃ = (M + IK×K)

and T̃ =(T + ρσS

σVM + ρσS

σVIK×K

). After some manipulation, the distortion Dlinear is given by

Dlinear =1

KEF[tr

{(T̃TFT [σ2S(1− ρ2)FM̃M̃TFT + σ2W IK×K ]−1FT̃ + σ−2V IK×K

)−1}]=

1

KEF[tr{(

QFTRF + σ−2V IK×K)−1}]

(13)

where we define Q = T̃T̃T , R = [σ2S(1−ρ2)FM̃M̃TFT +σ2W IK×K ]−1 and use the fact that for

any square matrices A and B, tr (I + AB)−1 = tr (I + BA)−1 [33]. Now by noting that for

any positive-definite K × K square matrix D, tr(D−1) ≥∑K

i=1 D−1ii [33], where Dii denotes

the diagonal elements in D and equality holds iff D is diagonal, we can write the following

Dlinear ≥1

K

K∑i=1

1

Qii|Fii|2Rii + σ−2V. (14)

Equality in (14) holds iff Q and R are diagonal; hence the optimal solution gives a diagonal T

and M. Thus, any linear coding can be achieved in a scalar form without performance loss.

C. Tandem Digital Scheme

In [34], Gel’fand and Pinsker showed that the capacity of a point-to-point communication

with side information (interference) known at the encoder side is given by

C = maxp(u,x|s)

I(U ;Y )− I(U ;S) (15)

where the maximum is over all joint distributions of the form p(s)p(u, x|s)p(y|x, s) and U

denotes an auxiliary random variable. In [18], Costa showed that using U = X + αS, with

α = PP+σ2W

over AWGN channel with interference known at the transmitter, the achievable

capacity is C = 12

log(

1 + Pσ2W

), which coincides with the capacity when both encoder and

decoder know the interference S. As a result, this choice of U is optimal in terms of maximizing

capacity. Next, we adapt the Costa scheme for the fading channel; we choose U = X + αS as


12

above, where α is redesigned to fit our problem. Using (15) and by interpreting the fading F as

a second channel output, an achievable rate R is given by

R = I(U ;Y, F )− I(U ;S) = I(U ;Y |F )− I(U ;S) (16)

where we used the fact that I(U ;F ) = 0. After some manipulations, the rate R is

R = EF[

1

2log

(P [f 2(P + σ2S) + σ

2W ]

Pσ2Sf2(1− α)2 + σ2W (P + α2σ2S)

)]. (17)

To find α, we minimize the expected value of the denominator in (17) (i.e., EF [Pσ2Sf 2(1 −

α)2 + σ2W (P + α2σ2S)]). As a result, we choose α =

PE[f2]PE[f2]+σ2W

for finite noise levels. Note that

this choice of α is independent of S and depends on the second order statistics of the fading.

In [35], the authors show that by choosing α = PP+σ2W

, Costa coding maximizes the achievable

rate for fading channels in the limits of both high and low noise levels.

The tandem scheme is based on the concatenation of an optimal source code and the adapted

Costa coding (described above). The optimal source code quantizes the analog source with a

rate close to that in (17), and the adapted Costa coder achieves a rate equal to (17). Hence, from

the lossy JSCC theorem, the MSE distortion for a K : N system can be expressed as follows

Dtandem =σ2V

exp{EF[r log

(P [f2(P+σ2S)+σ

2W ]

Pσ2Sf2(1−α)2+σ2W (P+α2σ

2S)

)]} (18)where r = N/K is the system’s rate. Note that the performance of this scheme does not improve

when the noise level decreases (levelling-off effect) or in the presence of correlation between

the source and the interference.

Remark 1 For the AWGN channel, the distortion of the tandem scheme in (18) can be simplified

as follows Dtandem = σ2V/

(1 + P/σ2W )r. This can be shown by setting α = P/(P + σ2W ) and

cancelling out the expectation in (18). This scheme is optimal for the uncorrelated case (ρ = 0).

IV. DISTORTION REGION FOR THE LAYERED SCHEMES

In this section, we propose layered schemes based on Wyner-Ziv and HDA coding for trans-

mitting a Gaussian source over a fading channel with correlated interference. These schemes

require proper combination of power splitting, bandwidth splitting, rate splitting, Wyner-Ziv and

HDA coding. A performance analysis in the presence of noise mismatch is also conducted.


13

A. Scheme 1: Layering Wyner-Ziv Costa and HDA for Bandwidth Expansion

This scheme comprises two layers that output XK1 and XN−K2 . The channel input is obtained by

multiplexing (concatenating) the output codeword of both layers XN = [XK1 XN−K2 ] as shown in

Fig. 2. The first layer is composed of two sublayers that are superimposed to produce the first K

samples of the channel input XK1 = XKa +X

Kd . The first sublayer is purely analog and consumes

an average power of Pa; the output of this sublayer is given by XKa =√a(β1V

K+β2SK), where

β1, β2 ∈ [−1 1], a = Paβ21σ2V +β22σ2S+2β1β2ρσV σS with 0 ≤ Pa ≤ P . The second sublayer, that outputs

XKd and consumes the remaining power Pd = P−Pa, encodes the source V K using a Wyner-Ziv

coder followed by a (generalized) Costa coder. The Wyner-Ziv encoder, which uses the fact that

an estimate of V K can be obtained at the decoder side, forms a random variable TK1 as follows

TK1 = αwz1VK +BK1 (19)

where each sample in BK1 is a zero mean i.i.d. Gaussian, αwz1 and the variance of B1 are defined

later. The encoding process starts by generating a K-length i.i.d. Gaussian codebook T1 of size

2KI(T1;V ) and randomly assigning the codewords into 2KR1 bins with R1 defined later. For each

source realization V K , the encoder searches for a codeword TK1 ∈ T1 such that (V K , TK1 ) are

jointly typical. In the case of success, the Wyner-Ziv encoder transmits the bin index of this

codeword using Costa coding. The Costa coder, which treats the analog sublayer XKa in addition

to SK as interference, forms the following auxiliary random variable UKc1 = XKd +αc1Š

K , where

ŠK = (XKa +SK), the samples in XKd are i.i.d. zero mean Gaussian with variance Pd = P −Pa

and 0 ≤ αc1 ≤ 1 is a real parameter. Note that XKd is independent of V K and SK . The encoding

process of the Costa coding can be summarized as follows

• Codebook Generation: Generate a K-length i.i.d. Gaussian codebook Uc1 with 2KI(Uc1 ;Y1,F )

codewords, where Y K1 is the first K samples of the received signal YN . Every codeword

is generated following the random variable UKc1 and uniformly distributed over 2KR1 bins.

The codebook is revealed to both encoder and decoder.

• Encoding: For a given bin index (the output of the Wyner-Ziv encoder), the Costa encoder

searches for a codeword UKc1 such that the bin index of UKc1

is equal to the Wyner-Ziv

output and (UKc1 , ŠK) are jointly typical. In the case of success, the Costa encoder outputs

XKd = UKc1− αc1ŠK . Otherwise, an encoding failure is declared. Note that the probability

of encoder failure vanishes by using R1 = I(Uc1 ;Y1, F )− I(Uc1 ; Š).


14

Wyner-ZivEncoder1

CostaEncoder1

β1

β2

+

+

√

a

XK1

XKa

XKdV K

SK

Wyner-ZivEncoder2

CostaEncoder2

Mux

XN−K2

[SN ]NK+1

XN

Fig. 2. Scheme 1 (bandwidth expansion) encoder structure.

The second layer, which outputs XN−K2 , encodes VK using a Wyner-Ziv with rate R2 and

a Costa coder that treats [SN ]NK+1 as interference. The Wyner-Ziv encoder, which uses the fact

that an estimate of V K is obtained from the first layer, forms the random variable TK2 as follows

TK2 = αwz2VK +BK2 (20)

where the samples in BK2 are i.i.d. and follow a zero mean Gaussian distribution, αwz2 and the

variance of B2 are defined later. The Costa coder forms the auxiliary random variable UN−Kc2 =

XN−K2 + αc2 [SN ]NK+1, where the samples in X

N−K2 are i.i.d. zero mean Gaussian with variance

P , and the real parameter αc2 is defined later. The encoding process of the Wyner-Ziv and the

Costa coder for the second layer is very similar to the one described for the first layer; hence,

no details are provided.

At the receiver side, as shown in Fig. 3, from the first K components of the received signal

Y N = [Y K1 , YN−K

2 ] = FN(XN + SN) + WN , where Y K1 = [Y

N ]K1 and YN−K

2 = [YN ]NK+1,

the Costa decoder estimates the codeword UKc1 by searching for a codeword UKc1

such that

(UKc1 , YK

1 , FK) are jointly typical. By the result of Gelfand-Pinsker [34] (or Costa [18]) and by

treating the fading coefficient FK as a second channel output, the error probability of encoding

and decoding the codeword UKc1 vanishes as K →∞ if

R1 = I(Uc1 ;Y1, F )− I(Uc1 ; Š) = I(Uc1 ;Y1|F )−(h(Uc1)− h(Uc1|Š)

)= h(Uc1) + h(Y1|F )− h(Uc1 , Y1|F )− h(Uc1) + h(Uc1 |Š)

= EF

[1

2log

(Pd[f

2(Pd + σ2Š) + σ2W ]

Pdσ2Šf2(1− αc1)2 + σ2W (Pd + α2c1σ

2Š)

)](21)


15

where σ2Š

= E[(Xa + S)2]. We then obtain a linear MMSE estimate of V K (based on Y K1 and

UKc1 ), denoted by VKa . The distortion from estimating the source using V

Ka is given by

Da = EF[σ2V − ΓΛ−1ΓT

](22)

where Λ = E[[Uc1 Y1]T [Uc1 Y1]] is the covariance of [Uc1 Y1] and Γ = E[V [Uc1 Y1]] is the

correlation vector between V and [Uc1 Y1]. By using rate R1 on the Wyner-Ziv encoder, the

bin index of the Wyner-Ziv can be decoded correctly (with high probability). The Wyner-Ziv

decoder then looks for a codeword TK1 in this bin such that (TK1 , V

Ka ) are jointly typical (as

K → ∞, the probability of error in decoding TK1 vanishes). A better estimate of V K is then

obtained based on V Ka and the decoded codeword TK1 . The distortion in the estimated source

Ṽ K is then

D̃ =Da

exp{EF[log(

Pd[f2(Pd+σ2Š

)+σ2W ]

Pdσ2Šf2(1−αc1 )2+σ

2W (Pd+α

2c1σ2Š

)

)]} . (23)Note that this distortion is equal to the distortion incurred when assuming that the side informa-

tion V Ka is also known at the transmitter side; this can be achieved by choosing αwz1 =√

1− D̃Da

and B1 ∼ N (0, D̃) in (19) and using a linear MMSE estimator based on V Ka and TK1 . In contrast

to the AWGN channel with correlated interference [26], a purely analog layer is not sufficient to

accommodate for the correlation over AWGN fading channel with correlated interference; indeed

using the knowledge of UKc1 as a side information to obtain a better description of the Wyner-Ziv

codewords TK1 will achieve a better performance. From the last N−K received symbols Y N−K2 ,

Wyner-Ziv

Decoder1

Costa

Decoder1

UK

c1

VK

a

ṼK

YN

Wyner-Ziv

Decoder2Costa

Decoder2

V̂K

LMMSEEstimator

Demux

UN−K

c2Y

N−K

2

YK

1

Fig. 3. Scheme 1 (bandwidth expansion) decoder structure.

the Costa decoder estimates the codeword UN−Kc2 by searching for a codeword UN−Kc2

such that

(UN−Kc2 , YN−K

2 , [FN ]NK+1) are jointly typical. The probability of error in encoding and decoding

the codeword UN−Kc2 goes to zero by choosing

R2 = I(Uc2 ;Y2, F )− I(Uc2 ;S) = EF[

1

2log

(P [f 2(P + σ2S) + σ

2W ]

Pσ2Sf2(1− αc2)2 + σ2W (P + α2c2σ

2S)

)](24)


16

where αc2 = PE[f 2]/(PE[f 2]+σ2W ) is found in a similar way as done in Sec. III-C. By using this

rate, the Wyner-Ziv bin index can be decoded correctly (with high probability). The Wyner-Ziv

decoder then looks for a codeword TK2 in the decoded bin such that TK2 and the side information

from the first layer Ṽ K are jointly typical. A refined estimate of the source can be found using

the side information Ṽ K and the decoded codeword TK2 . The resulting distortion is then

DScheme 1 = infβ1,β2,Pa,αc1

D̃

exp{EF[log(

P [f2(P+σ2S)+σ2W ]

Pσ2Sf2(1−αc2 )2+σ

2W (P+α

2c2σ2S)

)r−1]} . (25)

Note that this distortion is equal to the distortion realized when assuming Ṽ K is also known at

the transmitter side; this can be achieved using a linear MMSE estimator based on [T1 T2 Y1],

and by setting αwz2 =√

1− DScheme 1D̃

and B2 ∼ N (0, DScheme 1) in (20).

Remark 2 For AWGN channels with no fading, the same scheme can be used. In this case, the

distortion from reconstructing the source can be expressed as follows

DScheme 1 = infβ1,β2,Pa

{Da/

[(1 + P/σ2W )r−1(1 + Pd/σ

2W )]}. (26)

This distortion can be found by setting the fading coefficient F = 1, αc1 = Pd/(Pd + σ2W ) and

αc2 = P/(P +σ2W ) in (25). The distortion in (26) can be also achieved by replacing the sublayer

that outputs XKd by an HDA Costa layer as we proposed in [28]. Note that using only YK

1 as

input to the LMMSE estimator in Fig. 3 is enough for the AWGN case. In such case, Da in (26)

can be simplified as follows

Da =

(σ2V −

(√aβ21σ

2V + (

√aβ2 + 1)ρσV σS)

2

P + (2√aβ2 + 1)σ2S + 2

√aβ1ρσV σS + σ2W

). (27)

Moreover, one can check that this scheme is optimal (for the AWGN channel) for ρ = 0 and

ρ = 1. For ρ = 0, this happens by shutting down the analog sublayer (i.e., Pa = 0) in the scheme

and using (η1 = 1, η2 = 1) on the outer bound in (3). For the case of ρ = 1, the optimal power

allocation for the scheme is (Pa = P, Pd = 0). The resulting system’s distortion can be shown

to be equal to the outer bound in (3) for (η1 = 1, η2 = 0).

Scheme 1 under mismatch in noise levels: Next, we study the distortion of the proposed

scheme in the presence of noise mismatch between the transmitter and the receiver. The actual

channel noise power σ2Wa is assumed to be lower than the design one σ2W (i.e., σ

2Wa

< σ2W ).


17

Under such assumption, the Costa and Wyner-Ziv decoders are still able to decode correctly all

codewords with low probability of error. After decoding TK1 and TK2 , a symbol-by-symbol linear

MMSE estimator of V K based on Y K1 , TK1 and T

K2 is calculated. Hence Scheme 1’s distortion

under noise mismatch is D(Scheme 1)-mis = EF[σ2V − ΓTΛ−1Γ

], where Λ is the covariance matrix

of [T1 T2 Y1], and Γ is the correlation vector between V and [T1 T2 Y1]. Note that σ2Wa is

used in the covariance matrix Λ instead of σ2W .

Remark 3 When σ2Wa > σ2W , all codewords cannot be decoded correctly at the receiver side;

as a result we can only estimate the source vector V K by applying a linear MMSE estimator

based on the noisy received signal Y K1 . The system’s distortion in this case is given by

D(Scheme 1)-mis = EF[σ2V −

f 2(√aβ21σ

2V + (


2

f 2(P + (2√aβ2 + 1)σ2S + 2

√aβ1ρσV σS) + σ2Wa

]. (28)

B. Scheme 2: Layering Wyner-Ziv Costa and HDA for Bandwidth Reduction

In this section, we present a layered scheme for bandwidth reduction. This scheme comprises

three layers that are superposed to produce the channel input XN = XNa + XN1 + X

N2 , where

XNa , XN1 and X

N2 denote the outputs of the first, second and third layers, respectively. The

scheme’s encoder structure is depicted in Fig. 4. Recall that we denote the first N samples of

V K by V N and the last K − N samples by [V K ]KN+1. The first layer is an analog layer that

outputs XNa =√a(β1V

N +β2SN), a linear combination between the V N and SN , and consumes

Pa ≤ P as average power, where β1, β2 ∈ [−1 1], and a = Paβ21σ2V +β22σ2S+2β1β2ρσV σS is a gain factor

related to the power constraint Pa. The second layer, which operates on the first N samples

of the source, encodes V N using a Wyner-Ziv with rate R1 followed by a Costa coder. The

Wyner-Ziv encoder forms a random variable

TN1 = αwz1VN +BN1 (29)

where the samples in BN1 are i.i.d and follow a zero mean Gaussian distribution, the parameter

αwz1 and the variance of B1 are related to the side information from the first layer and hence

defined later. The Costa coder that treats both XNa and SN as interference forms the following

auxiliary random variable UNc1 = XN1 + αc1Š

N , where the samples in XN1 are i.i.d. zero mean

Gaussian with variance P1 ≤ P − Pa and independent of the source and the interference,

ŠN = XNa + SN and 0 ≤ αc1 ≤ 1 is a real parameter. The last layer encodes [V K ]KN+1 using


18

an optimal source encoder with rate R2 followed by a Costa coder. The Costa encoder, which

treats the outputs of the first two layers (XNa , XN1 ) as well as S

N as known interference, forms

the following auxiliary random variable UNc2 = XN2 + αc2S̃

N , where S̃N = (XNa + XN1 + S

N),

the samples in XN2 are zero mean i.i.d. Gaussian with variance P2 = P − P1 − Pa and αc2 =

P2E[f 2]/(P2E[f 2] + σ2W ).

Wyner-Ziv

Encoder1

Costa

Encoder1

β1

β2

+

+

√

aXN

a

XN1V K

SN

SourceEncoder2

CostaEncoder2

XN2

SN

XN

[V K ]KN+1

V N

Demux

Fig. 4. Scheme 2 (bandwidth reduction) encoder structure.

At the receiver, as shown in Fig. 5, from the received signal Y N the Costa decoder estimates

UNc1 . By using a rate R1 = I(Uc1 ;Y, F )−I(Uc1 ; Š) = EF[

12

log(

P1[f2(P1+σ2Š+P2)+σ2W ]

P1(σ2Š)f2(1−αc1 )2+(σ

2W+f

2P2)(P1+α2c1σ2Š

)

)],

where σ2Š

= E[(Xa + S)2], the Costa decoder (of the second layer) is able to estimate the

codewords UNc1 with vanishing error probability. We then obtain an estimate of VN , denoted by

V Na , using a linear MMSE estimator based on YN and UNc1 . The distortion from estimating V

N

using V Na is then given by

Da = EF[σ2V − ΓΛ−1ΓT

](30)

where Λ is the covariance of [Uc1 Y ] and Γ is the correlation vector between V and [Uc1 Y ].

The Wyner-Ziv decoder (of the second layer) then looks for a codeword TN1 such that (TN1 , V

Na )

are jointly typical (as N → ∞, the probability of error in decoding TN1 vanishes). A better

estimate of V N is then obtained based on the side information V Na and the decoded codeword

TN1 . The distortion from reconstructing VN is then given by

D1 =Da

exp(EF[log(

P1[f2(P1+σ2Š+P2)+σ2W ]

P1(σ2Š)f2(1−αc1 )2+(σ

2W+f

2P2)(P1+α2c1σ2Š

)

)]) . (31)Note that the distortion in (31) can be found by choosing αwz1 =

√1− D1

Daand B1 ∼ N (0, D1)

in (29) and using a linear MMSE estimator based on V Na and TN1 . To get an estimate of


19

[V K ]KN+1, we use a Costa decoder followed by a source decoder. Codewords of this layer can be

decoded correctly (with high probability) by choosing the rate R2 = I(Uc2 ;Y, F )− I(Uc2 ; S̃) =

EF[

12

log(

P2[f2(P2+σ2S̃

)+σ2W ]

P2σ2S̃f2(1−α2)2+σ2W (P2+α

22σ

2S̃

)

)], where σ2

S̃= E[(Xa + X1 + S)2]. The distortion in

reconstructing [V K ]KN+1 can be found by equating the rate-distortion function to the transmission

rate R2; this means that K−N2 logσ2VD2

= (N)R2. As a result, the distortion in reconstructing

[V K ]KN+1, denoted by D2, is given by

D2 =σ2V

exp{EF[

r1−r log

(P2[f2(P2+σ2

S̃)+σ2W ]

P2σ2S̃f2(1−α2)2+σ2W )(P2+α

22σ

2S̃

)

)]} . (32)Hence, the system’s distortion is given by DScheme 2 = infβ1,β2,Pa,P1,αc1

{rD1 + (1− r)D2

}.

Wyner-Ziv

Decoder1

Costa

Decoder1

UN

c1

VN

a

V̂N

YN

Source

Decoder2Costa

Decoder2 [V̂ K ]KN+1

LMMSEEstimator

UK−N

c2

V̂K

Mux

Fig. 5. Scheme 2 (bandwidth reduction) decoder structure.

Remark 4 For the AWGN channel, the distortion D1 and D2 for the reduction case are

D1 =Da

1 + P1P2+σ2W

and D2 =σ2V(

1 + P2σ2W

) r1−r

. (33)

Since for AWGN channel, the use of UNc1 as input to the LMMSE estimator in Fig. 5 does not

improve the performance, the distortion Da admits a simplified expression as given in (27).

The distortions in (33) can be derived by choosing αc1 =P1

P1+P2+σ2Wand αc2 =

P2P2+σ2W

. Note

that this scheme is optimal for uncorrelated source-interference and for full correlation between

the source and the interference. For the uncorrelated case, the analog layer is not needed

(Pa = 0, Da = σ2V ) and the optimal power allocation between the two other layers can be

derived by minimizing the resulting distortion with respect to P1; the optimal power P1 is

P ∗1 = σ2W

[1−

(1 + P

σ2W

)1−r]+P . For the case of full correlation between the (first N samples

of the) source and the interference (ρ = 1), the second layer can be shut down (P1 = 0) and

the optimal P ∗a satisfies

σ2W

(1 +

σW√Pa

)(1 +

P − Paσ2W

) 11−r(P + σ2W +

√Paσ2V

)−(P + σ2W + σ

2V + 2

√Paσ2V

)2= 0.


20

Scheme 2 under mismatch in noise levels: We next examine the distortion of the proposed

scheme in the presence of noise mismatch between the transmitter and the receiver. The actual

channel noise power σ2Wa is assumed to be lower than the design one σ2W (i.e., σ

2Wa

< σ2W ).

Under such assumption, the Costa and Wyner-Ziv decoders can decode all codewords with

vanishing probability of error. The distortion in reconstructing [V K ]KN+1, D2−mis, is hence the

same as in the matched noise level case; and the distortion from reconstructing V N is D1−mis =

EF[σ2V − ΓTΛ−1Γ

], where Λ is the covariance matrix of [T1 Y ], and Γ is the correlation vector

between V and [T1 Y ]. As a result, the system’s distortion is D(Scheme 2)-mis = rD1−mis + (1 −

r)D2−mis. Note that σ2Wa is used in Λ instead of σ2W when computing D1−mis.

Remark 5 When σ2Wa > σ2W , all codewords cannot be decoded correctly at the receiver side;

as a result we can only estimate the source vector V N by applying a linear MMSE estimator

based on the noisy received signal Y N . The system’s distortion is then given by

D(Scheme 2)-mis = rEF[(σ2V −

f 2(√aβ21σ

2V + (


2

f 2(P + (2√aβ2 + 1)σ2S + 2

√aβ1ρσV σS) + σ2Wa

)]+ (1− r)σ2V .

C. Numerical Results

In this section, we assume an i.i.d. zero-mean Gaussian source with unitary variance that is

transmitted over an AWGN Rayleigh fading channel with Gaussian interference. The interference

power is σ2S = 1, the power constraint is set to P = 1 and the Rayleigh fading has E[F 2] = 1. To

evaluate the performance, we consider the signal-to-distortion ratio (SDR = E[||V K ||2]/E[||V K−

V̂ K ||2]); the designed channel signal-to-noise ratio (CSNR , PE[F2]

σ2W) is set to 10 dB for all

numerical results. Fig. 6, which considers the AWGN channel, shows the SDR performance

versus the correlation coefficient ρ for bandwidth expansion (r = 2) and matched noise levels

between the transmitter and receiver. We note that the proposed scheme outperforms the tandem

Costa reference scheme (described in Sec. III-C) and performs very close to the “best” derived

outer bound for a wide range of correlation coefficients. Although not shown, the proposed

scheme also outperforms significantly the linear scheme of Sec. III-B. For the limiting cases of

ρ = 0 and 1, we can notice that the SDR performance of the proposed scheme coincides with

the outer bound and hence is optimal. Figs. 7, 8 and 9 show the SDR performance versus ρ

for the fading channel with interference under matched noise levels and for r = 1, 2 and 1/2,

respectively. As in the case of the AWGN channel, we remark that the proposed HDA schemes


21

0 0.2 0.4 0.6 0.8 120

21

22

23

24

25

26

27

28

29

ρ

SD

R [dB

]

Outer bound with η1=1 and η

2=1


2=0

Outer bound with optimized η1 and η

2

HDA Scheme 1

Tandem Costa scheme

Fig. 6. Performance of HDA Scheme 1 (r = 2) over the AWGN channel under matched noise levels for different correlation

coefficients. Tandem scheme and outer bounds on SDR are plotted. The graph is made for P = 1, σ2S = 1 and CSNR=10 dB.

outperform the tandem Costa and the linear schemes and perform close to the best outer bound.

In contrast to the AWGN case, the proposed scheme never coincides with the outer bound for

finite noise levels; this can be explained from the fact that the (generalized) Costa and linear

scheme are not optimal for the fading case. The sub-optimality (assuming that the outer bound

is tight) of the generalized Costa coding comes from the form of the auxiliary random variable.

We choose the same form as the one used for AWGN channels (a form linear in S); it remains

unclear if such auxiliary random variable is optimal for fading channels. Note that using the

result in [35], one can easily show that our schemes are optimal for ρ = 0 in the limits of

high and low noise levels. As a result, the auxiliary random variable used for the Costa coder

is optimal in the noise level limits.

Fig. 10 shows the SDR performance versus CSNR levels under mismatched noise levels. All

schemes in Fig. 10 are designed for CSNR=10 dB, r = 1 and ρ = 0.7. The true CSNR varies

between 0 and 35 dB. We observe that the proposed scheme is resilient to noise mismatch due

to its hybrid digital-analog nature. As the correlation coefficient values decreases, the power

allocated to the analog layer decreases. Hence, the SDR gap between the proposed and the

tandem Costa scheme under mismatched noise levels decreases and the robustness (which is the

trait of analog schemes) reduces.


22

0 0.2 0.4 0.6 0.8 18

10

12

14

16

18

ρ

SD

R [

dB

]

Outer bound with η1=η

2=1 and r=1

Outer bound with η1=0, η

2=1 and r=1


2 and r=1

HDA Scheme 1 with r=1

Linear scheme

Tandem Costa scheme for r=1

Fig. 7. Performance of Scheme 1 (r = 1) over the fading channel under matched noise levels for different correlation coefficient.

Tandem, linear schemes and outer bounds on SDR are plotted. The graph is made for P = 1, σ2S = 1, CSNR=10 dB and

E[F 2] = 1.

0 0.2 0.4 0.6 0.8 115

20

25

30

ρ

SD

R [dB

]


2=1


2=0


2

HDA Scheme 1 with r=2

Tandem Costa scheme

Fig. 8. Performance of Scheme 1 (r = 2) over the fading channel under matched noise levels for different correlation coefficient.

Tandem scheme and outer bounds on SDR are plotted. The graph is given for P = 1, σ2S = 1, CSNR=10 dB and E[F 2] = 1.

V. JSCC FOR SOURCE-CHANNEL-STATE TRANSMISSION

As an application of the joint source-channel coding problem examined in this paper we con-

sider the transmission of analog source-channel-state pairs over a fading channel with Gaussian

state interference. We establish inner and outer bounds on the source-interference distortion for

the fading channel. The only difference between this problem and that examined in the previous

sections is that the decoder is also interested in estimating the interference SN . For simplicity, we

focus on the matched bandwidth case (i.e., K = N ); the unequal source-channel bandwidth case

can be treated in a similar way as in Section IV. We also assume that the decoder has knowledge

of the fading. We denote the distortion from reconstructing the source and the interference by


23

0 0.2 0.4 0.6 0.8 12

3

4

5

6

7

8

9

10

11

ρ

SD

R [dB

]


2=1


2=0


2

HDA Scheme 2

Tandem Costa scheme

Fig. 9. Performance of Scheme 2 (r = 1/2) over the fading channel under matched noise levels for different ρ. Tandem scheme

and outer bounds on SDR are plotted. The graph is given for P = 1, σ2S = 1, CSNR=10 dB and E[F 2] = 1.

0 5 10 15 20 25 30 350

2

4

6

8

10

12

14

16

18

CSNR [dB]

SD

R [dB

]

HDA Scheme 1 (r=1, ρ=0.7)

Tandem Costa scheme (r=1, ρ=0.7)

Fig. 10. Performance of Scheme 1 (r = 1) over the fading channel under mismatched noise levels. Tandem scheme, analog

and upper bounds on SDR are plotted. The system is designed for P = 1, σ2S = 1, CSNR=10 dB, ρ = 0.7 and E[F 2] = 1.

Dv =1KE[||V K− V̂ K ||2] and Ds = 1KE[||S

K− ŜK ||2], respectively. For a given power constraint

P , a rate r and a Rayleigh fading channel, the distortion region is defined as the closure of all

distortion pair (D̃v, D̃s) for which (P, D̃v, D̃s) is achievable, where a power-distortion triple is

achievable if for any δv, δs > 0, there exist sufficiently large integers K and N with N/K = r,

encoding and decoding functions satisfying (2), such that Dv < D̃v + δv and Ds < D̃s + δs.

A. Outer Bound

Lemma 4 For the matched bandwidth case, the outer bound on the distortion region (Dv, Ds)

can be expressed as follows

Dv ≥Var(V |S)

exp{EF[log

ζP |f |2+σ2Wσ2W

]} , Ds ≥ σ2Sexp

{EF[log

|f |2(P+σ2S+2

√(1−ζ)Pσ2S

)+σ2W

ζP |f |2+σ2W

]} (34)September 12, 2014 DRAFT

24

where Var(V |S) = σ2V (1− ρ2) is the variance of V given S and 0 ≤ ζ ≤ 1.

Proof: For the source distortion, we can write the following

K

2log

σ2VDv

(a)

≤ I(V K ; V̂ K |FK)(b)

≤ I(V K ; V̂ K |FK) + I(V K ;SK |V̂ K , FK)

= I(V K ; V̂ K , SK |FK) (c)= I(V K ;SK) + I(V K ; V̂ K |SK , FK)(d)

≤ K2

logσ2V

Var(V |S)+ I(V K ;Y K |SK , FK)

=K

2log

σ2VVar(V |S)

+ h(Y K |SK , FK)− h(WK)

(e)=

K

2log

σ2VVar(V |S)

+K

2EF[log

ζP |f |2 + σ2Wσ2W

](35)

where (a) follows from the rate-distortion theorem, (b) follows from the non-negativity of

mutual information, (c) follows from the chain rule of mutual information and the fact that

FK is independent of (V K , SK), (d) holds by the data processing inequality and in (e) we used

h(Y K |SK , FK) = K2EF [log (ζP |f |2 + σ2W )] for some ζ ∈ [0, 1]; this can be proved from the fact

that K2

log σ2W = h(WN) ≤ h(Y K |SK , FK) ≤ h(FKXK+WK |FK) = K

2EF [log (P |f |2 + σ2W )].

Hence, there exists a ζ ∈ [0, 1] such that h(Y K |SK , FK) = K2EF [log (ζP |f |2 + σ2W )].

For the interference distortion, we have the following

K

2log

σ2SDs

(a)

≤ I(SK ; ŜK |FK)(b)

≤ I(SK ;Y K |FK) = h(Y K |FK)− h(Y K |SK , FK)

(c)

≤ supX∈B

EF[K

2log 2πe(|f |2(P + σ2S + 2E[SX]) + σ2W )

]−EF

[K

2log 2πe(ζP |f |2 + σ2W )

](d)= EF

[K

2log|f |2(P + σ2S + 2

√(1− ζ)Pσ2S) + σ2W

ζP |f |2 + σ2W

](36)

where (a) follows from the rate-distortion theorem, (b) follows from data processing inequality

for the mutual information, in (c) the set B = {X : h(Y K |SK , FK) = EF[K2

log 2πe(ζP |f |2 + σ2W )]}

and the inequality in (c) holds from the fact that Gaussian maximizes differential entropy and

h(Y K |SK , FK) = K2EF [log (ζP |f |2 + σ2W )] (as used in (35)). Note that the supremum over X

in (c) happens when Y , S and W are jointly Gaussian given F (i.e., X∗ is Gaussian). Now, we

represent X∗ = N∗ζ +X∗ζ , where N

∗ζ ∼ N (0, ζP ) is independent of X∗ζ ∼ N (0, (1− ζ)P ). Note

that X∗ζ is a function of S. Using this form of X∗, h(Y K |SK , FK) = K

2EF [log (ζP |f |2 + σ2W )]


25

still holds (i.e., X∗ ∈ B) and E[X∗S] = E[X∗ζS]. Maximizing over X is equivalent to maximizing

over E[XS]; using Cauchy-Schwarz(E[X∗S] = E[X∗ζS] ≤

√E[(X∗ζ )2]E[S2]

)we get (d).

B. Proposed Hybrid Coding Scheme

The proposed scheme is composed of three layers as shown in Fig. 11. The first layer, which

is purely analog, consumes an average power Pa and outputs a linear combination between

the source and the interference XKa =√a1(β11V

K + β12SK), where β11, β12 ∈ [−1 1] and

a1 =Pa

(β211σ2V +2β11β12ρσV σS+β

212σ

2S)

is a gain factor related to power constraint Pa. The second layer

employs a source-channel vector-quantizer (VQ) on the interference; the output of this layer is

XKq = µ(SK + UKq ), where µ > 0 is a gain related to the power constraint and samples in U

Kq

follow a zero mean i.i.d. Gaussian that is independent of V and S and has a variance Q. A

similar VQ encoder was used in [21] for the broadcast of bivariate sources and for the multiple

access channel [36]. In what follows, we outline the encoding process of the VQ

• Codebook Generation: Generate a K-length i.i.d. Gaussian codebook Xq with 2KRq code-

words with Rq defined later. Every codeword is generated following the random variable

XKq ; this codebook is revealed to both the encoder and decoders.

• Encoding: The encoder searches for a codeword XKq in the codebook that is jointly typical

with SK . In case of success, the transmitter sends XKq .

Wyner-Ziv

Encoder

Costa

Encoder

β11

β12

+

+

√

a1

XK

XKa

XKd

V K

SK

+

SK

β̃21

β̃22

X̃Kwz

VQXKq

Encoder

Fig. 11. Encoder structure.

The last layer encodes a linear combination between V K and SK , denoted by X̃Kwz, using a Wyner-

Ziv with rate R followed by a Costa coder. The Costa coder uses an average power of Pd and

treats XKa , SK and XKq as known interference. The linear combination X̃

Kwz = β̃21V

K+β̃22SK =

√a2(β21V

K +β22SK), where β21, β22 ∈ [−1 1] and a2 = Pd(β221σ2V +2β21β22ρσV σS+β222σ2S) . The Wyner-

Ziv encoder forms a random variable TK as follows


26

TK = αwzX̃Kwz +B

K (37)

where the samples in BK are zero mean i.i.d. Gaussian, the parameter αwz and the variance

of B are defined later. The encoding process of the Wyner-Ziv starts by generating a K-length

i.i.d. Gaussian codebook T of size 2KI(T ;X̃wz) and randomly assigning the codewords into 2KR

bins with R defined later. For each realization X̃Kwz, the Wyner-Ziv encoder searches for a

codeword TK ∈ T such that (X̃Kwz, TK) are jointly typical. In the case of success, the Wyner-

Ziv encoder transmits the bin index of this codeword using Costa coding. The Costa coder,

that treats S̃K = XKa +XKq + S

K as known interference, forms the following auxiliary random

variable UKc = XKd + αcS̃

K , where each sample in XKd is N (0, Pd) that is independent of the

source and the interference and 0 ≤ αc ≤ 1. The encoding process for the Costa coder can be

described in a similar way as done before.

At the receiver side, as shown in Fig. 12, from the noisy received signal Y K , the VQ decoder

estimates XKq by searching for a codeword XKq ∈ Xq that is jointly typical with the received

signal Y K and FK . Following the error analysis of [37], the error probability of decoding XKqgoes to zero by choosing the rate Rq to satisfy I(S;Xq) ≤ Rq ≤ I(Xq;Y, F ), where

I(S;Xq) = h(Xq)− h(Xq|S) =1

2log

σ2S +Q

Q

I(Xq;Y, F ) = I(Xq;F ) + I(Xq;Y |F ) = h(Y |F )− h(Y |Xq, F )

= EF{

1

2log 2πe

(E[Y 2]

)− 1

2log 2πe

(E[Y 2]− E[XqY ]

2

E[X2q ]

)}. (38)

The variance Q has to be chosen to satisfy the above rate constraint. Furthermore, to ensure the

power constraint is satisfied we need µ to satisfy Pa + µ2(σ2S +Q) + 2µE[SXa] + Pd ≤ P . The

Costa decoder then searches for a codeword UKc that is jointly typical with (YK , FK). Since

the received signal Y K and the codewords XKq and UKc are correlated with X̃

Kwz, an LMMSE

estimate of X̃Kwz, denoted byˆ̃XKwz, can be obtained based on Y

K and the decoded codewords

XKq and UKc . Mathematically, the estimate is given by

ˆ̃Xwz = ΓaΛ−1a [Xq Uc Y ]

T , where Λa is

the covariance of [Xq Uc Y ] and Γa is the correlation vector between X̃wz and [Xq Uc Y ]. The

distortion in reconstructing X̃Kwz is then Da = EF[Pd − ΓaΛ−1a ΓTa

]. Moreover, the Wyner-Ziv

decoder estimates the codeword TK by searching for a TK ∈ T that is jointly typical withˆ̃XKwz. The error probability of decoding both codewords T

K and UKc vanishes as K →∞ if the


27

coding rate of the Wyner-Ziv and the Costa coder is set to

R = EF

[1

2log

(Pd[f

2(Pd + σ2S̃) + σ2W ]

Pd(σ2S̃)f2(1− αc)2 + σ2W (Pd + α2cσ2S̃)

)](39)

where σ2S̃

= E[(Xa+Xq+S)2]. A better estimate of X̃Kwz can be obtained by using the codeword

TK and ˆ̃XKwz. The distortion in reconstructing X̃Kwz can be expressed as follows

D̃ =Da

exp{EF[log(

Pd[f2(Pd+σ2S̃

)+σ2W ]

Pd(σ2S̃

)f2(1−αc)2+σ2W (Pd+α2cσ2S̃

)

)]} . (40)This distortion can be achieved using a linear MMSE estimate based on TK , XKq and Y

K by

choosing αwz =√

1− D̃Da

and B ∼ N (0, D̃) in (37).

Wyner-Ziv

Decoder1

Costa

Decoder1

UKc

ˆ̃XKwz

TK

Y K

LMMSEEstimator

LMMSE

Estimator

(V̂ K , ŜK)

VQ

Decoder

XKq

Fig. 12. Decoder structure.

After decoding TK , XKq , a linear MMSE estimator is used to reconstruct the source and the

interference signals. As a result, the distortion in decoding V K and SK are given as follows

Dv = EF[σ2V − ΓvΛ−1ΓTv

]Ds = EF

[σ2S − ΓsΛ−1ΓTs

](41)

where Λ is the covariance of [Xq T Y ], Γv is the correlation vector between V and [Xq T Y ]

and Γs is the correlation vector between S and [Xq T Y ].

Remark 6 Using a linear combination of the source and the interference X̃Kwz instead of just

the source V K as an input to the Wyner-Ziv encoder in Fig. 11 is shown to be beneficial in some

parts of the source-interference distortion region. However, quantizing a linear combination of

the source and the interference by the VQ encoder (instead of just SK as done in Fig. 11) does

not seem to give any improvement.

Remark 7 For the AWGN channel with ρ = 0, using the source itself instead of X̃Kwz as input to

the Wyner-Ziv encoder, shutting down the second layer and setting β11 = 0 in XKa give the best


28

possible performance; the inner bound in such case coincides with the outer bound, hence the

scheme is optimal. This result is analogous to the optimality result of the rate-state-distortion for

the transmission of a finite discrete source over a Gaussian state interference derived in [29].

C. Numerical Results

We consider a source-interference pairs that are transmitted over a Rayleigh fading channel

(E[F 2] = 1) with Gaussian interference and power constraint P = 1; the CSNR level is set to

10 dB. For reference, we adapt the scheme of [29] to our scenario. Recall that the source and

the interference are jointly Gaussian, hence V K = ρσVσSSK +NKρ , where samples in N

Kρ are i.i.d.

Gaussian with variance σ2Nρ = (1− ρ2)σ2V and independent of S

K . Now if we quantize NKρ into

digital data, the setup becomes similar to the one considered in [29]; hence the encoding is done

by allocating a portion of the power, denoted by Ps, to transmit SK and the remaining power

Pd = (P − Ps) is used to communicate the digitized NKρ using the (generalized) Costa coder.

The received signal of such scheme is given by Y K = FK(√

Psσ2SSK +XKd + S

K)

+WK , where

XKd denotes the output of the digital part that communicates NKρ . An estimate of S

K is obtained

by applying a LMMSE estimator on the received signal; the distortion from reconstructing V K

is equal to the sum of the distortions from estimating ρσVσSSK and NKρ . Mathematically, the

distortion region of such reference scheme can be expressed as follows

Ds = EF[σ2S −

E[SY ]2

E[Y 2]

]= EF

[σ2S −

f 2(√PsσS + σ

2S)

2

f 2(P + σ2S + 2√PsσS) + σ2W

],

Dv = ρ2σ

2V

σ2SDs +

σ2Nρ

exp{EF[log(

Pd[f2(Pd+σ2S̃

)+σ2W ]

Pd(σ2S̃

)f2(1−αc)2+σ2W (Pd+α2cσ2S̃

)

)]} (42)where σ2

S̃= E[(

√Ps/σ2SS + S)

2] and the parameter αc is related to the Costa coder. To

evaluate the performance, we plot the outer bound (given by (34)) and the inner bounds (the

achievable distortion region) of the proposed hybrid coding (given by (41)) and the adapted

scheme of [29]. Fig. 13, which considers the AWGN channel, shows the distortion region of

the source-interference pair for ρ = 0.8 and σ2S = 0.5. We can notice that the hybrid coding

scheme is very close to the outer bound and outperforms the scheme of [29]. Moreover, the use

of the VQ layer is shown to be beneficial under certain system settings. Fig. 14, which considers

the fading channel, shows the distortion region of the source-interference pair for ρ = 0.8 and

σ2S = 1. The hybrid coding scheme performs relatively close to the outer bound.


29

−9 −8.5 −8 −7.5 −7 −6.5 −6−16.5

−16

−15.5

−15

−14.5

−14

−13.5

−13

−12.5

10 log10(Dv)

10log 1

0(D

s)

Adapted scheme of [29]

Hybrid coding without VQ layer

Hybrid coding

Outer bound

Fig. 13. Distortion region for hybrid coding scheme over the AWGN channel. This graph is made for σ2V = 1, P = 1, σ2S = 0.5

and ρ = 0.8.

−14 −12 −10 −8 −6 −4

−14

−12

−10

−8

−6

−4

10 log10 (Dv)

10log 1

0(D

s)

Adapted scheme of [29]

Hybrid coding scheme

Outer bound

Fig. 14. Distortion region for hybrid coding scheme over the fading channel. This graph is made for σ2V = 1, P = 1, σ2S = 1,

ρ = 0.8 and E[F 2] = 1.

VI. SUMMARY AND CONCLUSIONS

In this paper, we considered the problem of reliable transmission of a Gaussian sources over

Rayleigh fading channels with correlated interference under unequal source-channel bandwidth.

Inner and outer bounds on the system’s distortion are derived. The outer bound is derived by

assuming additional knowledge at the decoder side; while the inner bound is found by analyzing

the achievable distortion region of the proposed hybrid coding scheme. Numerical results show

that the proposed schemes perform close to the derived outer bound and to be robust to channel

noise mismatch. As an application of the proposed schemes, we derive inner and outer bounds

on the source-channel-state distortion region for the fading interference channel; in this case,


30

the receiver is interested in estimating both source and interference. Our setting contains several

interesting limiting cases. In the absence of fading and/or correlation and for some source-channel

bandwidths, our setting resorts to the scenarios considered in [20], [26], [29].

ACKNOWLEDGMENT

The authors sincerely thank the anonymous reviewers for their helpful comments.

REFERENCES

[1] C. E. Shannon, “A mathematical theory of communication,” The Bell System Technical Journal, vol. 27, pp. 379–423,

1948.

[2] T. A. Ramstad, “Shannon mappings for robust communication,” Telektronikk, vol. 98, no. 1, pp. 114–128, 2002.

[3] F. Hekland, P. A. Floor, and T. A. Ramstad, “Shannon-Kotel’nikov mappings in joint source-channel coding,” IEEE Trans.

Commun., vol. 57, no. 1, pp. 94–105, Jan 2009.

[4] J. (Karlsson) Kron and M. Skoglund, “Optimized low-delay source-channel-relay mappings,” IEEE Trans. Commun.,

vol. 58, no. 5, pp. 1397–1404, May 2010.

[5] P. A. Floor, A. N. Kim, N. Wernersson, T. Ramstad, M. Skoglund, and I. Balasingham, “Zero-delay joint source-channel

coding for a bivariate Gaussian on a Gaussian MAC,” IEEE Trans. Comm., vol. 60, no. 10, pp. 3091–3102, Oct 2012.

[6] Y. Hu, J. Garcia-Frias, and M. Lamarca, “Analog joint source channel coding using non-linear mappings and MMSE

decoding,” IEEE Trans. Commun., vol. 59, no. 11, pp. 3016–3026, Nov 2011.

[7] E. Akyol, K. Rose, and T. Ramstad, “Optimal mappings for joint source channel coding,” in IEEE Inform. Theory Works.,

Cairo, Egypt, 2010.

[8] E. Akyol, K. Rose, and K. Viswanatha, “On zero-delay source-channel coding,” IEEE Trans. Inform. Theory, in review.

[9] M. Kleiner and B. Rimoldi, “Asymptotically optimal joint source-channel coding with minimal delay,” in Proc. IEEE

Global Telecommunications Conference, Honolulu, HI, 2009.

[10] X. Chen and E. Tuncel, “Zero-delay joint source-channel coding using hybrid digital-analog schemes in the Wyner-Ziv

setting,” IEEE Trans. Communications, vol. 62, no. 2, pp. 726–735, Feb 2014.

[11] A. Abou Saleh, F. Alajaji, and W.-Y. Chan, “Power-constrained bandwidth-reduction source-channel mappings for fading

channels,” in Proc. of the 26th Bien. Symp. on Comm., Kingston, ON, May 2012.

[12] U. Mittal and N. Phamdo, “Hybrid digital analog (HDA) joint source channel codes for broadcasting and robust

communications,” IEEE Trans. Inform. Theory, vol. 48, pp. 1082–1102, May 2002.

[13] M. Skoglund, N. Phamdo, and F. Alajaji, “Hybrid digital-analog source-channel coding for bandwidth compres-

sion/expansion,” IEEE Trans. Inform. Theory, vol. 52, no. 8, pp. 3757–3763, Aug 2006.

[14] Y. Wang, F. Alajaji, and T. Linder, “Hybrid digital-analog coding with bandwidth compression for Gaussian source-channel

pairs,” IEEE Trans. Communications, vol. 57, no. 4, pp. 997–1012, Apr 2009.

[15] J. Nayak, E. Tuncel, and D. Gunduz, “Wyner-Ziv coding over broadcast channels: digital schemes,” IEEE Trans. Inform.

Theory, vol. 56, no. 4, pp. 1782–1799, Apr 2010.

[16] Y. Gao and E. Tuncel, “Wyner-Ziv coding over broadcast channels: Hybrid digital/analog schemes,” IEEE Trans. Inform.

Theory, vol. 57, no. 9, pp. 5660–5672, Sept 2011.

[17] E. Koken and E. Tuncel, “Gaussian HDA coding with bandwidth expansion and side information at the decoder,” in Proc.

IEEE Int. Symp. on Inform. Theory, Istanbul, Turkey, July 2013.


31

[18] M. Costa, “Writing on dirty paper,” IEEE Trans. Inform. Theory, vol. 29, no. 3, pp. 439–441, May 1983.

[19] M. P. Wilson, K. R. Narayanan, and G. Caire, “Joint source-channel coding with side information using hybrid digital

analog codes,” IEEE Trans. Inform. Theory, vol. 56, no. 10, pp. 4922–4940, Oct 2010.

[20] M. Varasteh and H. Behroozi, “Optimal HDA schemes for transmission of a Gaussian source over a Gaussian channel with

bandwidth compression in the presence of an interference,” IEEE Trans. Signal Processing, vol. 60, no. 4, pp. 2081–2085,

April 2012.

[21] C. Tian, S. N. Diggavi, and S. Shamai, “The achievable distortion region of sending a bivariate Gaussian source on the

Gaussian broadcast channel,” IEEE Trans. Inform. Theory, vol. 57, no. 10, pp. 6419–6427, Oct 2011.

[22] A. Lapidoth and S. Tinguely, “Broadcasting correlated Gaussians,” IEEE Trans. Inform. Theory, vol. 56, no. 7, pp. 3057–

3068, July 2010.

[23] H. Behroozi, F. Alajaji, and T. Linder, “On the performance of hybrid digital-analog coding for broadcasting correlated

Gaussian sources,” IEEE Trans. Commun., vol. 59, no. 12, pp. 3335–3342, Dec 2011.

[24] Y. Gao and E. Tuncel, “Separate source-channel coding for transmitting correlated Gaussian sources over degraded broadcast

channels,” IEEE Trans. Inform. Theory, vol. 59, no. 6, pp. 3619–3634, June 2013.

[25] A. Abou Saleh, F. Alajaji, and W.-Y. Chan, “Hybrid digital-analog coding for interference broadcast channels,” in Proc.

IEEE Int. Symp. on Inform. Theory, Istanbul, Turkey, July 2013.

[26] Y.-C. Huang and K. R. Narayanan, “Joint source-channel coding with correlated interference,” IEEE Trans. Commun.,

vol. 60, no. 5, pp. 1315–1327, May 2012.

[27] A. D. Wyner and J. Ziv, “The rate-distortion function for source coding with side information at the decoder,” IEEE Trans.

Information Theory, vol. 22, pp. 1–10, Jan 1976.

[28] A. Abou Saleh, W.-Y. Chan, and F. Alajaji, “Low and high-delay source-channel coding with bandwidth expansion and

correlated interference,” in Proc. of the 13th Canadian Workshop on Information Theory, Toronto, ON, June 2013.

[29] A. Sutivong, M. Chiang, T. Cover, and Y.-H. Kim, “Channel capacity and state estimation for state-dependent Gaussian

channels,” IEEE Trans. Inform. Theory, vol. 51, no. 4, pp. 1486–1495, Apr 2005.

[30] B. Bandemer, C. Tian, and S. Shamai, “Gaussian state amplification with noisy state observations,” in Proc. IEEE Int.

Symp. on Inform. Theory, Istanbul, Turkey, July 2013.

[31] A. Abou Saleh, F. Alajaji, and W.-Y. Chan, “Low-latency source-channel coding for fading channels with correlated

interference,” IEEE Wireless Communications Letter, vol. 3, no. 2, pp. 137–140, Apr 2014.

[32] Y.-K. Chia, R. Soundararajan, and T. Weissman, “Estimation with a helper who knows the interference,” in Proc. IEEE

Int. Symp. on Inform. Theory, Cambridge, MA, July 2012.

[33] K. M. Abadir and J. R. Magnus, Matrix Algebra. New York: cambridge University Pressr, 2005.

[34] S. I. Gelfand and M. S. Pinsker, “Coding for channel with random parameters,” Probl. Inform. Contr., vol. 9, no. 1, pp.

19–31, 1980.

[35] W. Zhang, S. Kotagiri, and J. Laneman, “Writing on dirty paper with resizing and its application to quasi-static fading

broadcast channels,” in Proc. IEEE Int. Symp. on Inform. Theory, Nice, France, June 2007.

[36] A. Lapidoth and S. Tinguely, “Sending a bivariate Gaussian over a Gaussian MAC,” IEEE Trans. Information Theory,

vol. 56, no. 6, pp. 2714–2752, 2010.

[37] S. H. Lim, P. Minero, and Y.-H. Kim, “Lossy communication of correlated sources over multiple access channels,” in

Proceedings of the 48th Annual Allerton Conf. on Comm., Control and Computing, Monticello, IL, Sep 2010.


Documents

1 Source-Channel Coding for Fading Channels with Correlated …communications/Papers/fdpc-tcom.pdf · 2014. 9. 26. · 1 Source-Channel Coding for Fading Channels with Correlated